integral (a^2-x^2)^{1/2}

Lösung

integral

(a^{2} - x^{2})^{\frac{1}{2}}

x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

Schritte zur Lösung

\int (a^{2} - x^{2})^{\frac{1}{2}} d x

Wende die partielle Integration an

= x (a^{2} - x^{2})^{\frac{1}{2}} - \int - \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} d x

\int - \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} d x = - \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

= x (a^{2} - x^{2})^{\frac{1}{2}} - (- \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))))

Vereinfache

= x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

Füge eine Konstante zur Lösung hinzu

= x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

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